Direct Integration of the Topological String

We present a new method to solve the holomorphic anomaly equations governing the free energies of type B topological strings. The method is based on direct integration with respect to the non–holomorphic dependence of the amplitudes, and relies on the interplay between non–holomorphicity and modularity properties of the topological string amplitudes. We develop a formalism valid for any Calabi–Yau manifold and we study in detail two examples, providing closed expressions for the amplitudes at low genus, as well as a discussion of the boundary conditions that fix the holomorphic ambiguity. The first example is the non-compact Calabi–Yau underlying Seiberg–Witten theory and its gravitational corrections. The second example is the Enriques Calabi–Yau, which we solve in full generality up to genus six. We discuss various aspects of this model: we obtain a new method to generate holomorphic automorphic forms on the Enriques moduli space, we write down a new product formula for the fiber amplitudes at all genus, and we analyze in detail the field theory limit. This allows us to uncover the modularity properties of SU(2), N = 2 super Yang–Mills theory with four massless hypermultiplets.